Magnetic Mixing - Michael A. Beauregard

Microfluidic devices consist of channels that are of dimensions that make most liquids, when flowed through these channels, flow in a laminar pattern. Achieving mixing in a microfluidic channel can have a plethora of applications including increasing rates of reaction or dilution and culturing cells. Magnetic mixing can accomplish these goals through three main resources: singular stir bars or beads, groups of magnetic beads, and ferrofluids.

Singular Stir Bars and Beads

Stir bars can range widely in size. Larger stir bars usually need to be incorporated into a larger, macroscopic container called a continuously stirred tank reactor (CSTR). The spinning motion of the bar or bead is accomplished using an external magnetic field. The effluent of the macroscopic reactor can then be fed into a microfluidic device for further experimentation and analysis. CSTRs follow the design equation below:^[1]

[math]\displaystyle{ C(t) = C_o e^{-F t/V} }[/math]

Where:

[math]\displaystyle{ C(t) = Concentration at Time t }[/math]

[math]\displaystyle{ C_o = Initial concentration }[/math]

[math]\displaystyle{ F = Flow Rate }[/math]

[math]\displaystyle{ V = CSTR Volume }[/math]

Figure 1. Macroscopic CSTR with black stir bead.Reproduced under the terms of the Creative Common CC-BY license. ^[2] Copyright 2015, The Authors. Journal of Chemical Technology & Biotechnology published by John Wiley & Sons Ltd on behalf of Society of Chemical Industry.

Meaningful changes in concentration with respect to time in microfluidic channels require small flow rates and consequently small CSTR volumes according to the design equation. CSTRs in conjunction with a microfluidic device often require a volume on the scale of ~1 mL. Tan, et al. characterize the use of different CSTR shapes of 1 mL volume and show that they can be used for cell cultures on a small scale (see Figure 1).^[2]

Stir bars can also be shrunk down to ~500 µm in length and ~30 µm in width.^[3] This allows them to fit inside microfluidic channels (see Figure 2). The stir bar is attached to an axel at its center. The axel is sandwiched between a glass base layer and a PDMS top layer so that the stir bar does not move when liquid is flowed into the channel. An external magnetic field must be strong enough to overcome the frictional forces of liquid moving through the channel to spin the stir bar. The stir bar is located at the interface of two liquids downstream of a Y junction where the two liquids meet (see Figure 3). Due to laminar flow, only areas in the channel that come into direct contact with the mixing bar become mixed. Liquid at the very edges of the channel continue to rely purely on diffusion for mixing.

• 

  Figure 2. Micro stir bar (left) and its placement into a microfluidic device (right).^[3]

• 

  Figure 3. Flow pattern resulting from a micro stir bar^[3]

Magnetic Beads

Magnetic beads can be used to achieve thorough mixing in very small spaces.^[4] Lee et al. developed a method where 4 µm magnetic beads are flowed into a 4.2 nL compartment of a microfluidic device. The beads remain in the mixing chamber due to a combination of their size and an externally applied magnetic field. Smaller beads would flow out of the mixing compartment and larger beads would conglomerate into a single mass that a magnetic field could not move. The correct beads would align and rotate just as a macroscopic stir bar would. When the beads were not rotating, they provided a mixing efficiency of 66%. When the beads were moving, they provided a mixing efficiency of 96%. The beads were also biocompatible.

• 

  Figure 4. Flow pattern resulting from the mixing of 4 µm diameter beads inside a CSTR (center) at increasing mixing rates from a-c.^[4]

Ferrofluids

Ferrofluids contain solid magnetic particles similar the beads discussed above, but they are ~10 nm in diameter, giving them properties more like a liquid when in an aqueous suspension.^[5] Wen et al. used a microfluidic technique where a channel of biocompatible ferrofluid met a channel of dye via a Y intersection (see Figure 5).^[6] The channel parallels a coil of wire. Different extents of mixing can be seen as time progresses after introducing a magnetic field (see Figure 6). Different extents of mixing can also depend on the frequency of the alternating current flowing through the wire, the strength of the resulting magnetic field, and the amount of time that the microfluidic channel spends next to the wire coil.

• 

  Figure 5. Ferrofluid suspension and a non-ferric solution enter on the left and come together at the Y junction. The fluids pass an electromagnet before exiting on the right.^[6]

• 

  Figure 6. Each photo was taken at the midpoint of the electromagnet at t = 0s, 1s, 3s, 5s, and 8s after the electromagnet was turned on.^[6]

References

1. Hill, C. G.; Root, T. W. An Introduction to Chemical Engineering Kinetics & Reactor Design, Second edition.; John Wiley & Sons, Inc: Hoboken, New Jersey, 2014.
2. Tan, C. K.; Davies, M. J.; McCluskey, D. K.; Munro, I. R.; Nweke, M. C.; Tracey, M. C.; Szita, N. Electromagnetic Stirring in a Microbioreactor with Non-Conventional Chamber Morphology and Implementation of Multiplexed Mixing: Electromagnetic Stirring in a Microbioreactor. J. Chem. Technol. Biotechnol. 2015, 90 (10), 1927–1936. https://doi.org/10.1002/jctb.4762.
3. Liang-Hsuan Lu; Kee Suk Ryu; Chang Liu. A Magnetic Microstirrer and Array for Microfluidic Mixing. J. Microelectromech. Syst. 2002, 11 (5), 462–469. https://doi.org/10.1109/JMEMS.2002.802899.
4. Lee, S. H.; van Noort, D.; Lee, J. Y.; Zhang, B.-T.; Park, T. H. Effective Mixing in a Microfluidic Chip Using Magnetic Particles. Lab Chip 2009, 9 (3), 479–482. https://doi.org/10.1039/B814371D.
5. Ferrofluids. https://education.mrsec.wisc.edu/ferrofluids-background/ (accessed Feb 28, 2022).
6. Wen, C.-Y.; Yeh, C.-P.; Tsai, C.-H.; Fu, L.-M. Rapid Magnetic Microfluidic Mixer Utilizing AC Electromagnetic Field. Electrophoresis 2009, 30 (24), 4179–4186. https://doi.org/10.1002/elps.200900400.
7. Microfluidic Technologies for Miniaturized Analysis Systems; Hardt, S., Schönfeld, F., Eds.; MEMS reference shelf; Springer: New York, NY, 2007.